# 趋于“零”就要考虑趋于“零负”和趋于“零正”两种情况

## 准备工作

$$1 – \cos ( a x ) = \textcolor{yellow}{ 2 \sin ^ { 2 } \left( \frac{a x}{2} \right) }$$

## 当 $a$ $>$ $0$ 时

\begin{aligned} I \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}}} \frac { x } { \sqrt { 1 – \cos ( a x ) } } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { x } { \sqrt { 2 \sin ^ { 2 } \left( \frac{a x}{2} \right) } } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { – x } { \sqrt { 2 } \sin \left( \frac{a x}{2} \right) } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { – x } { \sqrt { 2 } \frac{a x}{2} } \\ \\ & = \frac{-2}{\sqrt{2} a} \\ \\ & = \textcolor{green}{\boldsymbol{\frac { -\sqrt { 2 } } { a }}} \end{aligned}

\begin{aligned} I \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}}} \frac { x } { \sqrt { 1 – \cos ( a x ) } } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { x } { \sqrt { 2 \sin ^ { 2 } \left( \frac{a x}{2} \right) } } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { + x } { \sqrt { 2 } \sin \left( \frac{a x}{2} \right) } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { + x } { \sqrt { 2 } \frac{a x}{2} } \\ \\ & = \frac{+2}{\sqrt{2} a} \\ \\ & = \textcolor{green}{\boldsymbol{\frac { \sqrt { 2 } } { a }}} \end{aligned}

## 当 $a$ $<$ $0$ 时

\begin{aligned} I \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { x } { \sqrt { 1 – \cos ( a x ) } } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { x } { \sqrt { 2 \sin ^ { 2 } \left( \frac{a x}{2} \right) } } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { x } { \sqrt { 2 } \sin \left( \frac{a x}{2} \right) } \\ \\ & = \lim _{ \textcolor{orangered}{x \rightarrow 0^{-}} } \frac { x } { \sqrt { 2 } \frac{a x}{2} } \\ \\ & = \frac{2}{\sqrt{2} a} \\ \\ & = \textcolor{green}{\boldsymbol{\frac { \sqrt { 2 } } { a } }} \end{aligned}

\begin{aligned} I \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { x } { \sqrt { 1 – \cos ( a x ) } } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { x } { \sqrt { 2 \sin ^ { 2 } \left( \frac{a x}{2} \right) } } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { -x } { \sqrt { 2 } \sin \left( \frac{a x}{2} \right) } \\ \\ & = \lim _{ \textcolor{springgreen}{x \rightarrow 0^{+}} } \frac { -x } { \sqrt { 2 } \frac{a x}{2} } \\ \\ & = \frac{-2}{\sqrt{2} a} \\ \\ & = \textcolor{green}{\boldsymbol{\frac { -\sqrt { 2 } } { a } }} \end{aligned}